In this study, membranes of collagen–chitosan (C-Ch) in combination with conductive polymers (CPs) such as polyaniline (Pani) and polypyrrole (Ppy) were obtained by electrospinning using non-toxic solvents such as PBS and ethanol. The change in the morphology after swelling was observed by SEM, while an FTIR analysis showed specific interactions between C-Ch and CP. Mechanical tests showed that C-Ch/Ppy exhibited more elastic behavior and a better stress distribution compared to C-Ch/Pani. The diffusion of Na⁺ and Ca²⁺ ions through the membranes was evaluated and showed a greater resistance for Ca²⁺ in both membrane types. Preliminary biocompatibility testing with H9C2 cells showed a successful cell adhesion to the membranes. These results emphasize the potential of C-Ch/Pani composites for electrically active scaffolds and of C-Ch/PPy composites for applications in mechanically dynamic tissue-specific regeneration.